# Multi-factor model for a quantitative response variable

I am analyzing data in order to develop a model for power generation of a solar system based on three predictor variables. I have done t-tests showing significant improvement in the generation while varying one variable at a time. Each variable has only two possible values in my situation, and for each variable I have a 95% confidence interval on the change in generation.

While it is great to understand the effect of each variable alone, I would like to somehow develop a model that can predict the change in generation given the affects of multiple variables.

I'll write this out symbolically if that helps explain my situation.

Three predictor variables A B C

Each has two possible values, 1 or 0.

My response variable is G

I know how G changes when I vary A whilst B and C are held constant. I know the same for when I vary B and C (holding the other two constant).

I would like to know how G changes when I vary more than one of A, B, or C.

What sort of statistical process would I use to create such a model?

Comment to rolando2:

After reading your article, I am not quite sure that i'm working with interactions. Let me put numbers to what I have right now to make this more concrete.

One of the predictor variables is angle. I know that when changing the angle from 15 to 30 degrees, everything else being constant, I generate 80 more watts on average. This I found using a paired 2-sample t-test with daily data over a year for both angles side-by-side.

Another variable is inverter type. I know that when I use inverter B instead of inverter A, I produce 40 more watts on average.

I am guessing that if we put together a system with 30 degree tilt and inverter B, and compare that to our "standard" system with 15 degree tilt and inverter A, we would not generate 120 more watts on average.

How would I know how much more I would produce without having to experimentally test each combination?

P.S. My experience in multivariable statistics is very limited, as you probably might have guessed. I have a semester course of statistics based in R, following a year of AP Stats in HS.

-

I still agree with @rolando2. Let me start off with some stats 101 points for general information:

1. you use the tag for binary-data, however I gather your response variable is the amount of power generated, and what's binary are your predictors; statistical models rarely make assumptions about the predictors, so you needn't worry about that.
2. Multivariate is mostly used to denote situations where there is >1 response variable, that makes life a little more complicated. However, you appear to have 1 response variable, and multiple predictors, which is not as advanced, so again, not something you need to worry about.
3. You refer to 'prediction', but it's not clear to me from your description of your situation that what you really want is to be able to predict values in the future. It seems to me that you want to know what settings will maximize your power output. If that's true, you want to know about causal factors, and want to build an explanatory model. Moreover, you need to run an experiment.

Do you have experimental data for power generation where you've varied these 'predictors' simultaneously & (hopefully) orthogonally? If not, you need to do some runs to get that data. As I say, I agree with @rolando2 that what you are asking about are interaction effects. Thus, you will need to run a full factorial experiment to compute those effects; so far as I can tell based on what you've written, you cannot simply predict those effects based on what you've got so far. (Sorry to be the bearer of bad news--for what it's worth, you may have the data you need already, I just can't tell.)

Some information about these topics will probably be of value to you. I have found the online StatSoft electronic textbook to be helpful. You may want to read some parts from several topics: ANOVA, the general linear model, and possibly the design of experiments (note that the chapters are long, you needn't read everything). Hope that helps.

-
I have observational data from several systems, but I don't have observations for all combinations of my predictor variables. I wish I did, and I don't have the ability to set up and run more experiments. My goal is to try and do the best work with what I have. Of the eight possible combinations, I have five. At this point, I will read more into interactions, but I doubt that I will be able to do anything with the data I have. Thanks for all the help! –  Lukas Woltjer Jul 30 '12 at 13:09
What combinations do you have? You won't be able to assess the effects of the combinations you don't have, but you may be able to make some preliminary inferences. –  gung Jul 30 '12 at 13:11
I have all 4 combinations of module type and angle, but only with one type of inverter. The other inverter is the only one of it's sort in our system. –  Lukas Woltjer Jul 30 '12 at 13:36
In that case, you can examine the 2-way interaction b/t module & angle, but not the interactions b/t mod & inverter or angle & inverter, nor the 3-way interaction. You would need to gather data on the remaining combinations to determine the rest. –  gung Jul 30 '12 at 13:39
Alright, I see where we're going now. One last question: how would you recommend I examine this interaction? I have daily generation values over a year, which I have been using to determine the difference of generation between angles with module type held constant with a paired t-test. Would I use each of these daily values for the interaction, or just use the means over a year? –  Lukas Woltjer Jul 30 '12 at 14:02
show 1 more comment

It sounds as if you are talking about investigating statistical interactions, a.k.a. joint or multiplicative or moderator effects. You can read up on that and see what is the best way to implement such a model in your preferred software. Essentially, you'll want to test the predictive power of A*B, A*C, and B*C above and beyond what the main effects for A, B, and C can predict. If I have read you correctly in that this is your interest and that you are new to this sort of technique, then before you get into more detailed sources you might benefit from a very quick conceptual introduction I wrote, at YellowBrickStats.com.

-
If i'm not mistaken, this would be an appropriate guide. Also, because I have binomial predictors, I would simply use 1 and 0 for them, instead of the continuous data the tutorial has. –  Lukas Woltjer Jul 31 '12 at 14:27